Past exam of the mathematics course of the University of Cambridge / 2017 / ii / Paper 3 / 24J / a / Solution

A family $(X_n)$ of integrable random variables is uniformly integrable whenEquivalently the integrals of their tails vanish uniformly. This includes uniform $L^1$ boundedness, since $\mathbb{E}|X_n|\le M+\mathbb{E}[|X_n|1_{|X_n|>M}]$.